Example 9.21 from Prealgebra, 9.3 Use Properties of Angles, Triangles, and the Pythagorean Theorem

Use Properties of Angles, Triangles, and the Pythagorean Theorem: example 9.21

$Δ A B C$ and $Δ X Y Z$ are similar triangles. The lengths of two sides of each triangle are shown. Find the lengths of the third side of each triangle.

Two triangles are shown. They appear to be the same shape, but the triangle on the right is smaller. The vertices of the triangle on the left are labeled A, B, and C. The side across from A is labeled a, the side across from B is labeled 3.2, and the side across from C is labeled 4. The vertices of the triangle on the right are labeled X, Y, and Z. The side across from X is labeled 4.5, the side across from Y is labeled y, and the side across from Z is labeled 3.

Work it out on paper first. Then open the solution one step at a time, and stop as soon as you can finish on your own.

Your turn

The book's Try It right after this example: same idea, new numbers. Only the answer is given.

$Δ A B C$ is similar to $Δ X Y Z .$ Find $a .$

Two triangles are shown. They appear to be the same shape, but the triangle on the right is larger The vertices of the triangle on the left are labeled A, B, and C. The side across from A is labeled a, the side across from B is labeled 15, and the side across from C is labeled 17. The vertices of the triangle on the right are labeled X, Y, and Z. The side across from X is labeled 12, the side across from Y is labeled y, and the side across from Z is labeled 25.5.

How did it go?

Step 1. Read the problem. Draw the figure and label it with the given information.	The figure is provided.
Step 2. Identify what you are looking for.	The length of the sides of similar triangles
Step 3. Name. Choose a variable to represent it.	Let a = length of the third side of $Δ A B C$ y = length of the third side $Δ X Y Z$
Step 4. Translate.
The triangles are similar, so the corresponding sides are in the same ratio. So $\frac{A B}{X Y} = \frac{B C}{Y Z} = \frac{A C}{X Z}$ Since the side $A B = 4$ corresponds to the side $X Y = 3$ , we will use the ratio $\frac{AB}{XY} = \frac{4}{3}$ to find the other sides. Be careful to match up corresponding sides correctly.
Step 5. Solve the equation.
Step 6. Check:
Step 7. Answer the question.	The third side of $Δ A B C$ is 6 and the third side of $Δ X Y Z$ is 2.4.

Step 1. Read the problem. Draw the figure and label it with the given information.	The figure is provided.
Step 2. Identify what you are looking for.	The length of the sides of similar triangles
Step 3. Name. Choose a variable to represent it.	Let a = length of the third side of $Δ A B C$ y = length of the third side $Δ X Y Z$
Step 4. Translate.
The triangles are similar, so the corresponding sides are in the same ratio. So $\frac{A B}{X Y} = \frac{B C}{Y Z} = \frac{A C}{X Z}$ Since the side $A B = 4$ corresponds to the side $X Y = 3$ , we will use the ratio $\frac{AB}{XY} = \frac{4}{3}$ to find the other sides. Be careful to match up corresponding sides correctly.
Step 5. Solve the equation.
Step 6. Check:
Step 7. Answer the question.	The third side of $Δ A B C$ is 6 and the third side of $Δ X Y Z$ is 2.4.

Use Properties of Angles, Triangles, and the Pythagorean Theorem: example 9.21

Solution

Your turn

Use Properties of Angles, Triangles, and the Pythagorean Theorem: example 9.21

Solution

Your turn